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en:berechnungen:festigkeitsberechnung [2025/02/24 10:46] – [Shear Stress] cschallen:berechnungen:festigkeitsberechnung [2025/09/03 12:27] (aktuell) – [Shear Stress] neelest
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 ======Stress Analysis====== ======Stress Analysis======
  
--> [[grafische_darstellung_der_ergebnisse:spannungsanalyse|]]+-> [[..:grafische_darstellung_der_ergebnisse:spannungsanalyse|For graphical representation of the strength calculation]]
  
 ===== Equivalent Stress===== ===== Equivalent Stress=====
Zeile 9: Zeile 9:
 $$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$ $$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$
  
-Since the rotation of the screw is purely a torsion process, the hypothesis can be simplified by considering only the shear stress resulting from torsion. Additionally, bending of the screw can be neglected, meaning the normal stress in the y-direction can also be disregarded. The normal stress in the x-direction is determined solely by the pressures at the beginning and end of the screw.+Since the rotation of the screw represents a purely torsional load, the hypothesis can be simplified and only the shear stress resulting from torsion needs to be considered. 
 +As bending of the screw can also be neglected, the normal stress in the y-direction  
 +$\sigma_y$​ may likewise be disregarded. 
 +The normal stress in the x-direction $\sigma_x$ therefore results solely from the pressures at the inlet and outlet of the screw.
  
 $$σ_v = \sqrt {σ_x^2 + 4τ_{xy}^2}$$ $$σ_v = \sqrt {σ_x^2 + 4τ_{xy}^2}$$
Zeile 32: Zeile 35:
 The shear stress resulting from the applied torque is calculated using the following formula: The shear stress resulting from the applied torque is calculated using the following formula:
  
-$$\tau_{nenn} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{I_p}$$+$$\tau_{nominal} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{I_p}$$
  
 with the torque $M_t$, the section modulus $W_t$, the polar moment of inertia $I_p$ and the maximum perpendicular distance from the outer fiber to the neutral (stress-free) fiber $a_{max}$. with the torque $M_t$, the section modulus $W_t$, the polar moment of inertia $I_p$ and the maximum perpendicular distance from the outer fiber to the neutral (stress-free) fiber $a_{max}$.
  
-Another influencing factor for calculating the worm gear strength is the radius that describes the transition from the worm root to the web. For this geometric influence, a shape factor (stepped round bar under torsion, DIN 743-2) is defined as:+Another influencing factor for calculating the screw strength is the radius that describes the transition from the screw core to the flight. For this geometric influence, a shape factor (stepped round bar under torsion, DIN 743-2) is defined as:
  
 $$\alpha_{\tau} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$ $$\alpha_{\tau} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
Zeile 45: Zeile 48:
  
 $$\beta_{\tau} = \frac{\alpha_\tau}{n}$$ $$\beta_{\tau} = \frac{\alpha_\tau}{n}$$
-$$\text{mit}$$+$$\text{with}$$
 $$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$ $$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$
-$$\text{und}$$+$$\text{and}$$
 $$G'=\frac{1,15}{r}$$ $$G'=\frac{1,15}{r}$$
  
Zeile 54: Zeile 57:
 $$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$ $$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$
 $$\text{with}$$ $$\text{with}$$
-$$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{     width     } K_2(d>150\,mm)=0,8$$+$$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{     with     } K_2(d>150\,mm)=0,8$$
  
-with the influence of the surface roughness $K_{F,\tau}=1$ and the influence of surface hardening $K_V = 1,15$ for nitrided surfaces.+as well as with the influence of surface roughness $K_{F,\tau}=1$ for polished surfaces and the influence of surface hardening $K_V$. $K_V$ ranges from 1.15 to 1.25 for nitrided surfaces, 1.2 to 2.1 for case-hardened surfaces, and 1.2 to 1.6 for induction-hardened surfaces. 
 +The values apply up to a diameter of 25 mm and decrease linearly beyond this value down to 1.0 at a diameter of 40 mm. For diameters above 40 mm, the value remains constant at 1.0.
  
-Finally, the shear stress, taking all influence factors into account, is given by:+The resulting shear stress, taking into account these influencing factors, is given by:
  
-$$\tau_{xy} = \tau_{max}=\tau_{nenn} \cdot K_\tau$$+$$\tau_{xy} = \tau_{max}=\tau_{nominal} \cdot K_\tau$$
  
 ===Further topics=== ===Further topics===
   * [[en:berechnungen:einfache_berechnung|]]   * [[en:berechnungen:einfache_berechnung|]]
 +  * [[en:berechnungen:prozess_iterieren]]
   * [[en:berechnungen:durchsatz|]]   * [[en:berechnungen:durchsatz|]]
   * [[en:berechnungen:druckverlauf|]]   * [[en:berechnungen:druckverlauf|]]
Zeile 69: Zeile 74:
   * [[en:berechnungen:temperaturverlauf|]]   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]   * [[en:berechnungen:leistung_und_schubspannungen|]]
 +  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]   * [[en:berechnungen:verweilzeitverteilung|]]